Trying for a serious-ish answer here, folks!

I think it is probably a combination of several factors mentioned above (60% of Scots names..., relatively informal etc) together with two more: a lot of broadside ballads had place names written as J----, or R---- and the ides, I gather, was that you substituted something local. In the basis of no supporting evidence whatever, I would not be surprised if someoe knew of 'John' and 'Willy' being used as 'placeholders' in the same way.

Another notable thing about Will/Willy/William is that is equally acceptable with one, two or three syllables and the first two of these have lots of rhymes (as pointed out above). This must make life easier for both the songwriter and a singer with an imperfect memory!